Averkov, Gennadiy; Düvelmeyer, Nico Embedding metric spaces into normed spaces and estimates of metric capacity. (English) Zbl 1127.52006 Monatsh. Math. 152, No. 3, 197-206 (2007). Summary: Let \({\mathcal M}^d\) be an arbitrary real normed space of finite dimension \(d \geq 2\). We define the metric capacity of \({\mathcal M}^d\) as the maximal \(m \in {\mathbb N}\) such that every \(m\)-point metric space is isometric to some subset of \({\mathcal M}^d\) (with metric induced by \({\mathcal M}^d\)). We obtain that the metric capacity of \({\mathcal M}^d\) lies in the range from 3 to \(\left\lfloor\frac{3}{2}d\right\rfloor+1\), where the lower bound is sharp for all \(d\), and the upper bound is shown to be sharp for \(d \in \{2, 3\}\). Thus, the unknown sharp upper bound is asymptotically linear, since it lies in the range from \(d + 2\) to \(\left\lfloor\frac{3}{2}d\right\rfloor+1\). Cited in 2 Documents MSC: 52A21 Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry) 52B10 Three-dimensional polytopes 52C10 Erdős problems and related topics of discrete geometry Keywords:normed space; Banach space; metric space; hexagonal prism; two-distance set; parallelotope; maximum norm; \(l_\infty\)-space; rectilinear plane; isometric embedding PDFBibTeX XMLCite \textit{G. Averkov} and \textit{N. Düvelmeyer}, Monatsh. Math. 152, No. 3, 197--206 (2007; Zbl 1127.52006) Full Text: DOI References: [5] Düvelmeyer N (2006) Selected Problems from Minkowski Geometry. PhD thesis, Chemnitz Univ of Technology [8] Linial N (2002) Finite metric-spaces – combinatorics, geometry and algorithms. In: Li TT et al (eds) Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), pp 573–586. Beijing: Higher Education Press · Zbl 0997.05019 [21] Wernicke B (1994) Triangles and Reuleaux triangles in Banach-Minkowski planes. In: Böröczky K et al (eds) Intuitive Geometry (Szeged, 1991), Colloq Math Soc János Bolyai, Vol. 63, pp 505–511. Amsterdam: North-Holland · Zbl 0824.51016 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.